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Must scale a stirling engine type beta, and must show the calculations for dimensions. Where to start ?? Please give me tips

Maybe I have expressed myself badly. Sorry my english .... I'm doing a job for the construction of a beta type Stirling and do not know where to begin to calculate the prototype measurements.

Without Schmidt theory, designed with the basic steps:

cylinder diameter: 45mm
cylinder length: 350mm

With the information given I can only give you one dimension, the displacer should be 135 mm long, it will require 1 mm to 1.5 mm clearance in the gap, and at the ends. Anything else requires you to select the stroke required. The ratio between the power piston swept volume, and the swept volume of the displacer should be as near to 1.5 : 1.
The power piston area requires a very accurate bore.
You will have at least 100mm of spare tube left out of the 350 mm you have.
Ian S C